Discontinuous payoff functions have attracted some attention in the literature, especially in economic games. This paper considers such functions when the location of the discontinuity is known incompletely. Under this scenario, the presence of a discontinuity creates risk for economic agents. Standard intuition does not hold perfectly here: for instance, we show that an agent's optimal choice is a U-shaped function of the amount of risk, which is most surprising. We show that these results apply to many economic problems including issues of mechanism design, double auctions and standard monopoly pricing models.